Summary of Green’s Theorem

Green’s theorem relates the integral over a connected region to an integral over the boundary of the region. Green’s theorem is a version of the Fundamental Theorem of Calculus in one higher dimension.
Green’s Theorem comes in two forms: a circulation form and a flux form. In the circulation form, the integrand is [latex]{\bf{F}}\cdot{\bf{T}}[/latex]. In the flux form, the integrand is [latex]{\bf{F}}\cdot{\bf{N}}[/latex].
Green’s theorem can be used to transform a difficult line integral into an easier double integral, or to transform a difficult double integral into an easier line integral.
A vector field is source free if it has a stream function. The flux of a source-free vector field across a closed curve is zero, just as the circulation of a conservative vector field across a closed curve is zero.

Key Equations

Green’s theorem, circulation form
[latex]\displaystyle\oint_{C} Pdx+Qdy=\displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex], where [latex]C[/latex] is the boundary of [latex]D[/latex]
Green’s theorem, flux form
[latex]\displaystyle\oint_{C} {\bf{F}}\cdot {d{\bf{r}}} = \displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex], where [latex]C[/latex] is the boundary of [latex]D[/latex]
Green’s theorem, extended version
[latex]\displaystyle\oint_{\partial{D}} {\bf{F}}\cdot {d{\bf{r}}} = \displaystyle\iint_{D} Q_{x}-P_{y}dA[/latex]

Green’s theorem: relates the integral over a connected region to an integral over the boundary of the region

stream function: if [latex]{\bf{F}} = {\langle}P, Q{\rangle}[/latex] is a source-free vector field, then stream function g is a function such that [latex]P = g_{y}[/latex], and [latex]Q = -{g_{x}}[/latex]